nLab
Lie algebra

Idea

A Lie algebra is the infinitesimal approximation to a Lie group.

Definition

Ordinary definition

A Lie algebra is a vector space $𝔤$ equipped with a bilinear skew-symmetric map $[-,-{]}_{𝔤}\vee 𝔤\to 𝔤$ which satisfies the Jacobi identity:

$$\forall x,y,z\in 𝔤:[x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0.$$\forall x,y,z \in \mathfrak{g} : [x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0 \,.

A homomorphism of Lie algebras is a linear map $\varphi :𝔤\to 𝔥$ such that for all $x,y\in 𝔤$ we have

$$\varphi ([x,y{]}_{𝔤})=[\varphi (x),\varphi (y){]}_{𝔥}.$$\phi([x,y]_{\mathfrak{g}}) = [\phi(x),\phi(y)]_{\mathfrak{h}} \,.

This defines the category LieAlg of Lie algebras.

In a general linear category

The notion of Lie algebra may be formulated internal to any linear category. This general definition subsumes as special case generalizations such as super Lie algebras.

Given a commutative unital ring $k$, and a (strict for simplicity) symmetric monoidal $k$-linear category $(C,\otimes ,1)$ with the symmetry $\tau$, a Lie algebra in $(C,\otimes ,1,\tau )$ is an object $L$ in $C$ together with a morphism $[,]:A\otimes A\to A$ such that the Jacobi identity

$$[,[,]]+[,[,]]\circ ({\mathrm{id}}_L\otimes {\tau }_{L,L})\circ (\tau \otimes {\mathrm{id}}_L)+[,[,]]\circ ({\tau }_{L,L}\otimes {\mathrm{id}}_L)\circ ({\mathrm{id}}_L\otimes {\tau }_{L,L})=0$$[,[,]]+[,[,]]\circ(id_L\otimes\tau_{L,L})\circ(\tau\otimes id_L)+[,[,]]\circ (\tau_{L,L}\otimes id_L)\circ (id_L\otimes\tau_{L,L}) = 0

and antisymmetry

$$[,]+[,]\otimes {\tau }_{L,L}=0$$[,]+[,]\otimes\tau_{L,L} = 0

hold. If $k$ is the ring $\mathbb{Z}$ of integers, then we say (internal) Lie ring, and if $k$ is a field and $C=\mathrm{Vec}$ then we say a Lie $k$-algebra. Other interesting cases are super-Lie algebras, which are the Lie algebras in the symmetric monoidal category ${\mathbb{Z}}_2-\mathrm{Vec}$ of supervector spaces and the Lie algebras in the Loday-Pirashvili tensor category of linear maps.

Alternatively, Lie algebras are the algebras over certain quadratic operad, called the Lie operad, which is the Koszul dual of the commutative algebra operad.

Cohomology

See Lie algebra cohomology.

Lie theory

See

• Lie theory

  • Lie integration

  • Lie's three theorems

Extra stuff, structure, properties

Notions of Lie algebras with extra stuff, structure, property includes

• extra property

  • abelian Lie algebra

  • simple Lie algebra

  • semisimple Lie algebra

  • reductive Lie algebra

• extra structure

  • super Lie algebra

• extra stuff

  • ∞-Lie algebra

    • simplicial Lie algebra

  • ∞-Lie algebroid

Properties

• PBW theorem

Examples

• line Lie algebra

• general linear Lie algebra

• orthogonal Lie algebra

• special orthogonal Lie algebra

• special unitary Lie algebra

• Poincare Lie algebra

• super Poincare Lie algebra

Related concepts

• Lie algebra

  • universal enveloping algebra, Cartan subalgebra

  • root (in representation theory), weight (in representation theory)

  • Lie algebra representation

  • Kac-Moody algebra

  • Lie group

  • Lie-Poisson structure

• Lie 2-algebra, Lie 2-group

• L-∞ algebra, smooth ∞-group

• Lie algebroid, Lie groupoid

• L-∞ algebroid, smooth ∞-groupoid

Examples of sequences of infinitesimal and local structures

first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
	$\leftarrow$ differentiation		integration $\to$
derivative		Taylor series		germ		smooth function
tangent vector		jet		germ of curve		curve
Lie algebra		formal group		local Lie group		Lie group
Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization

References

For instance chapter IV in vol III of

• Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)